Dictionary of mathematics terms

PROFESSIONAL GUIDES Dictionary of Mathematics Terms Third Edition • More than 800 terms related to algebra, geometry, analytic geometry, trigonometry, probability, statistics, logic, and calculus • An ideal reference for math students, teachers, engineers, and statisticians • Filled with illustrative diagrams and a quick-reference formula summary Douglas Downing, Ph.D Dictionary of Mathematics Terms Third Edition Dictionary of Mathematics Terms Third Edition Douglas Downing, Ph.D School of Business and Economics Seattle Pacific University Dedication This book is for Lori Acknowledgments Deepest thanks to Michael Covington, Jeffrey Clark, and Robert Downing for their special help © Copyright 2009 by Barron’s Educational Series, Inc Prior editions © copyright 1995, 1987 All rights reserved No part of this publication may be reproduced or distributed in any form or by any means without the written permission of the copyright owner All inquiries should be addressed to: Barron’s Educational Series, Inc 250 Wireless Boulevard Hauppauge, New York 11788 www.barrronseduc.com ISBN-13: 978-0-7641-4139-3 ISBN-10: 0-7641-4139-2 Library of Congress Control Number: 2008931689 PRINTED IN CHINA CONTENTS Preface vi List of Symbols ix Mathematics Terms Appendix 381 Algebra Summary 381 Geometry Summary 382 Trigonometry Summary 384 Brief Table of Integrals 388 PREFACE Mathematics consists of rigorous abstract reasoning At first, it can be intimidating; but learning about math can help you appreciate its great practical usefulness and even its beauty—both for the visual appeal of geometric forms and the concise elegance of symbolic formulas expressing complicated ideas Imagine that you are to build a bridge, or a radio, or a bookcase In each case you should plan first, before beginning to build In the process of planning you will develop an abstract model of the finished object—and when you that, you are doing mathematics The purpose of this book is to collect in one place reference information that is valuable for students of mathematics and for persons with careers that use math The book covers mathematics that is studied in high school and the early years of college These are some of the general subjects that are included (along with a list of a few entries containing information that could help you get started on that subject): Arithmetic: the properties of numbers and the four basic operations: addition, subtraction, multiplication, division (See also number, exponent, and logarithm.) Algebra: the first step to abstract symbolic reasoning In algebra we study operations on symbols (usually letters) that stand for numbers This makes it possible to develop many general results It also saves work because it is possible to derive symbolic formulas that will work for whatever numbers you put in; this saves you from having to derive the solution again each time you change the numbers (See also equation, binomial theorem, quadratic equation, polynomial, and complex number.) Geometry: the study of shapes Geometry has great visual appeal, and it is also important because it is an vi vii example of a rigorous logical system where theorems are proved on the basis of postulates and previously proved theorems (See also pi, triangle, polygon, and polyhedron.) Analytic Geometry: where algebra and geometry come together as algebraic formulas are used to describe geometric shapes (See also conic sections.) Trigonometry: the study of triangles, but also much more Trigonometry focuses on six functions defined in terms of the sides of right angles (sine, cosine, tangent, secant, cosecant, cotangent) but then it takes many surprising turns For example, oscillating phenomena such as pendulums, springs, water waves, light waves, sound waves, and electronic circuits can all be described in terms of trigonometric functions If you program a computer to picture an object on the screen, and you wish to rotate it to view it from a different angle, you will use trigonometry to calculate the rotated position (See also angle, rotation, and spherical trigonometry.) Calculus: the study of rates of change, and much more Begin by asking these questions: how much does one value change when another value changes? How fast does an object move? How steep is a slope? These problems can be solved by calculating the derivative, which also allows you to answer the question: what is the highest or lowest value? Reverse this process to calculate an integral, and something amazing happens: integrals can also be used to calculate areas, volumes, arc lengths, and other quantities A first course in calculus typically covers the calculus of one variable; this book also includes some topics in multi-variable calculus, such as partial derivatives and double integrals (See also differential equation.) Probability and Statistics: the study of chance phenomena, and how that study can be applied to the analysis of data (See also hypothesis testing and regression.) viii Logic: the study of reasoning (See also Boolean algebra.) Matrices and vectors: See vector to learn about quantities that have both magnitude and direction; see matrix to learn how a table of numbers can be used to find the solution to an equation system with many variables A few advanced topics are briefly mentioned because you might run into certain words and wonder what they mean, such as calculus of variations, tensor, and Maxwell’s equations In addition, several mathematicians who have made major contributons throughout history are included The Appendix includes some formulas from algebra, geometry, and trigonometry, as well as a table of integrals Demonstrations of important theorems, such as the Pythagorean theorem and the quadratic formula, are included Many entries contain cross references indicating where to find background information or further applications of the topic A list of symbols at the beginning of the book helps the reader identify unfamiliar symbols Douglas Downing, Ph.D Seattle, Washington 2009 APPENDIX a b c d a b c a b c 382 ϭ ad bc ϭ a bc ϭ ac b Quadratic Formula If ax2 ϩ bx ϩ c ϭ 0, then x ϭ Ϫb Ϯ 2b2 Ϫ 4ac 2a GEOMETRY SUMMARY Plane Figures Triangles • sum of angles ϭ 180° • (area) ϭ 12 ϫ (base) ϫ (height) • If the lengths of the sides are a, b, and c, and s ϭ (a ϩ b ϩ c) / 2, the area is 2s1s Ϫ a21s Ϫ b21s Ϫ c2 • Pythagorean theorem for a right triangle: a2 ϩ b2 ϭ c2, where c is the hypotenuse Quadrilaterals • sum of angles ϭ 360° • square: a ϭ length of side; (area) ϭ a2; four 90° angles • rectangle: a and b are lengths of two adjacent sides; (area) ϭ ab; four 90° angles • paralellogram or rhombus: (area) ϭ (base) ϫ (height) 383 APPENDIX • trapezoid: a and b are lengths of two parallel sides; h is distance between those two sides; (area) ϭ h(a ϩ b)/2 Polygons • sum of angles for an n-sided polygon: 180 ϫ (nϪ 2) Regular Polygons • area of regular polygon with n sides inscribed in circle of radius r: 2p 1area2 ϭ nr2 sin a b n • area of regular polygon with n sides of length a: na2 sin a 2p b n 1area2 ϭ 2p Ϫ 4cos a b n Circle • r ϭ radius; (circumference) ϭ 2pr • (area) ϭ pr2 • area of sector of circle with angle u (radians): ur2 • area of segment of circle with angle u (radians): r2 1u Ϫ sinu2 Solid Figures Cube (side of length a) • (volume) ϭ a3 • (surface area) ϭ 6a2 APPENDIX 384 Sphere, radius r • (volume) ϭ 43pr3 • (surface area) ϭ 4pr2 Prism or Cylinder • (volume) ϭ (base area) ϫ (height) Cone or Pyramid • (volume) ϭ 13 ϫ (base area) ϫ (height) TRIGONOMETRY SUMMARY Trigonometric Functions for Right Triangles Let A be one of the acute angles in a right triangle Then: 1opposite side2 1hypotenuse2 1adjacent side2 cosA ϭ 1hypotenuse2 1oppositeside2 tan A ϭ 1adjacentside2 sinA ϭ Trigonometric Functions: General Definition Consider a point (x, y) in a Cartesian coordinate system Let r be the distance from that point to the origin, and let A be the angle between the x-axis and the line connecting the origin to that point Then: y sinA ϭ r x cos A ϭ r y tanA ϭ x 385 APPENDIX Radian Measure p rad ϭ 180° Special Values Degrees Radians sin cos tan 0° 30° 0 p 23 13 45° p 12 12 60° p 23 2 23 90° p Undefined (infinite) Trigonometric Identities These equations are true for all allowable values of A and B Reciprocal functions: sinA ϭ cscA cscA ϭ sinA cosA ϭ secA secA ϭ cos A tanA ϭ ctn A ctnA ϭ tanA Cofunctions (radian form): sinA ϭ cos a p Ϫ Ab cos A ϭ sin a p Ϫ Ab p Ϫ Ab ctnA ϭ tan a p Ϫ Ab p Ϫ Ab cscA ϭ sec a p Ϫ Ab tanA ϭ ctn a secA ϭ csc a APPENDIX 386 Negative angle relations: sin(ϪA) ϭϪ sin A cos(ϪA) ϭ cos A tan(ϪA) ϭϪ tan A Quotient relations: sinA cos A cos A ctnA ϭ sinA tanA ϭ Supplementary angle relations: The angles A and B are supplementary angles if A ϩ B ϭ p sin(␲Ϫ A) ϭ sin A cos(␲Ϫ A) ϭϪcos A tan(␲Ϫ A) ϭϪtan A Pythagorean identities: sin2 A ϩ cos2 A ϭ tan2 A ϩ ϭ sec2 A ctn2 A ϩ ϭ csc2 A Functions of the sum of two angles: sin(A ϩ B) ϭ sin A cos B ϩ sin B cos A cos(A ϩ B) ϭ cos A cos B Ϫ sin A sin B tanA ϩ tanB tan 1A ϩ B2 ϭ Ϫ tanAtanB Functions of the difference of two angles: sin(A Ϫ B) ϭ sin A cos B Ϫ sin B cos A cos(A Ϫ B) ϭ cos A cos B ϩ sin A sin B tanA Ϫ tan B tan 1A Ϫ B2 ϭ ϩ tanAtanB 387 APPENDIX Double-angle formulas: sin 12A2 ϭ 2sinAcos A cos 12A2 ϭ cos 2A Ϫ sin 2A ϭ Ϫ 2sin 2A ϭ 2cos 2A Ϫ tan 12A2 ϭ 2tanA Ϫ tan 2A Squared formulas: Ϫ cos 2A ϩ cos 2A cos 2A ϭ Half-angle formulas: sin 2A ϭ sin a A Ϫ cosA bϭ Ϯ B cos a A ϩ cosA bϭ Ϯ B tan a A Ϫ cosA bϭ Ϯ B ϩ cosA Product formulas: sin 1A ϩ B2 ϩ sin 1A Ϫ B2 sin 1A ϩ B2 Ϫ sin 1A Ϫ B2 cos AsinB ϭ cos 1A ϩ B2 ϩ cos 1A Ϫ B2 cosAcosB ϭ cos 1A ϩ B2 Ϫ cos 1A Ϫ B2 sinAsinB ϭ Ϫ sinAcosB ϭ APPENDIX 388 Sum formulas: AϩB AϪB b cos a b 2 AϩB AϪB cosA ϩ cosB ϭ 2cos a b cos a b 2 sinA ϩ sinB ϭ 2sin a ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Difference formulas: AϩB AϪB b sin a b 2 AϩB AϪB cosA Ϫ cos B ϭ Ϫ2sin a b sin a b 2 sin A Ϫ sinB ϭ 2cos a ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Formulas for triangles: Let a be the side of a triangle opposite angle A, let b be the side opposite angle B, and let c be the side opposite angle C Law of cosines: c2 ϭ a2 ϩ b2 Ϫ 2ab cos C Law of sines: a b c ϭ ϭ sinA sinB sinC BRIEF TABLE OF INTEGRALS a, b, c, m, n represent constants; C represents the arbitrary constant of integration Perfect Integral Ύ dx ϭ x ϩ C 389 APPENDIX Multiplication by Constant Ύ n dx ϭ nx ϩ C Ύ nf1x2 dx ϭ n Ύ f1x2 dx Ύ f1nx2dx ϭ n Ύ f1u2 du where u ϭ nx Addition Ύ ΄f1x2 ϩ g1x2΅dx ϭ Ύ f1x2 dx ϩ Ύ g1x2 dx Powers Ύ x dx ϭ ϩ C x2 Ύ x dx ϭ n ϩ ϩ C if n n Ύx Ϫ1 xnϩ1 Ϫ1 dx ϭ ln 0x ϩ C Polynomials Ύ 1a x n ϭ n ϩ anϪ1xnϪ1 ϩ p ϩ a2x2 ϩ a1x ϩ a0 2dx anxnϩ1 anϪ1xn a2x3 ϩ ϩ p ϩ n nϩ1 ϩ a1x2 ϩ a0x ϩ C APPENDIX 390 Substitution Ύ f1u1x2 2dx ϭ Ύ f1u2 dudu dx For example: Ύ xf1x ϩ a2 dx ϭ ϭ Ύ xf1u2 a 2x b du Ύ f1u2 du where u ϭ x2 ϩ a Integration by Parts Ύ udv ϭ uv Ϫ Ύ v du Note: The arbitrary constant of integration C will not be explicitly listed in the integrals that follow, but it must always be remembered Trigonometry Ύ sinx dx ϭ Ϫcos x Ύ cosx dx ϭ sinx Ύ tanx dx ϭ ln 0sec x Ύ sec x dx ϭ ln 0sec x ϩ tan x Ύ sin x dx ϭ Ϫ x sin2x Ύ x sin x dx ϭ sinx Ϫ xcosx 391 APPENDIX Ύ x sin x dx ϭ Ϫx cosx ϩ 2xsinx ϩ 2cosx 2 Ύ cos x dx ϭ ϩ x sin2x Ύ xcos x dx ϭ cos x ϩ xsinx Ύ sin x cos x dx ϭ Ύ sin 2x sin m x dx ϭ Ϫ sin mϪ1 xcos x m ϩ mϪ1 sin mϪ2x dx m Ύ Ύ arcsinx dx ϭ xarcsinx ϩ 21 Ϫ x Ύ arctanx dx ϭ xarctanx Ϫ ln 11 ϩ x2 2 Exponential Functions and Logarithms Ύ e dx ϭ e x x Ύ xe dx ϭ xe x x Ύ x e dx ϭ x e x Ύ Ύ ax dx ϭ ex cosx dx ϭ Ϫ ex x Ϫ 2xex ϩ 2ex ax lna ex sinx ϩ cos x2 APPENDIX 392 Ύ lnx dx ϭ x lnx Ϫ x Ύ Ύ Ύ x lnx dx ϭ x2 lnx x2 Ϫ x2 lnx dx ϭ x3 lnx x3 Ϫ q eϪx >2dx ϭ Ϫq 22p Integrals involving ax2 ؉ bx ؉ c For this section, let D ϭ b2 Ϫ 4ac These integrals can be simplified by substituting u ϭ x ϩ b/2a: ax2 ϩ bx ϩ c ϭ (1) Let y ϭ Ύ ax dx ϩ bx ϩ c If D Ͻ : y ϭ 2ϪD If D Ͼ : y ϭ 2D 4a2u2 Ϫ D 4a arctan a ln ` 2ax ϩ b 2ϪD 2ax ϩ b Ϫ 2D 2ax ϩ b ϩ 2D If D ϭ : y ϭ Ϫ 2ax ϩ b Specific examples of form (1) include: Ύ1 ϩ x dx ϭ arctanx Ύ1 Ϫ x 1ϩx dx ϭ ln ` ` 1Ϫx 2 b ` 393 APPENDIX Ύm 1 nx dx ϭ arctan a b mn m ϩ n2x2 Ύm 1 m ϩ nx dx ϭ ln ` ` m Ϫ nx 2mn Ϫ n2x2 2 Let y ϭ (2) If a Ͼ : y ϭ 2a Ύ 2ax ϩ bx ϩ c dx ln 022a1ax2 ϩ bx ϩ c2 ϩ 2ax ϩ b If a Ͻ and D Ͼ : y ϭ Ϫ1 2Ϫa arcsin a 2ax ϩ b 2D (provided 02ax ϩ b Ͻ 2D2 Specific examples of form (2) include: Ύ 21 Ϫ x dx ϭ arcsinx Ύ 21 ϩ x dx ϭ ln 1x ϩ 21 ϩ x2 Ύ 2x dx ϭ ln 1x ϩ 2x2 Ϫ 12 2 Ύ 2m Ϫ1 Ϫnx 2 Ύ 2n x 2 ϩ m2 dx ϭ nx arcsin a b n m dx ϭ nx n2x2 ln ϩ 1ϩ 2 n m B m b APPENDIX 394 Let y ϭ (3) yϭ a Ύ 2ax ϩ bx ϩ c dx 2ax ϩ b 2ax2 ϩ bx ϩ c ϩ 4a 4ac Ϫ b2 b 8a Ύ 2ax ϩ bx ϩ c dx Specific examples of form (3) include: Ύ 21 Ϫ x2 dx ϭ Ύ 21 ϩ x dx ϭ Ύ arcsinx ϩ x21 Ϫ x2 x21 ϩ x2 ϩ ln 0x ϩ 21 ϩ x2 x2x2 Ϫ Ϫ ln 0x ϩ 2x2 Ϫ 2x Ϫ dx ϭ 2 Ύ 2m2 Ϫ n2x2 dx ϭ m2 nx c arcsin a b m 2n ϩ Ύ 2m ϩ n2x2 dx ϭ nx nx 1Ϫ a b d m B m m2 nx nx ca b 1ϩ a b m B m 2n ϩ ln2 nx nx ϩ 1ϩ a b 2d m m B PROFESSIONAL GUIDES [...]... APOTHEM The apothem of a regular polygon is the distance from the center of the polygon to one of the sides of the polygon, in the direction that is perpendicular to that side ARC An arc of a circle is the set of points on the circle that lie in the interior of a particular central angle Therefore an arc is a part of a circle The degree measure of an arc is the same as the degree measure of the angle that... parabola y ϭ x2 is symmetric about the line x ϭ 0 (See axis of symmetry.) AXIS OF SYMMETRY An axis of symmetry is a line that passes through a figure in such a way that the part of the figure on one side of the line is the mirror image of the part of the figure on the other side of the line (See reflection.) For example, an ellipse has two axes of symmetry: the major axis and the minor axis (See ellipse.)... thermometers in which the temperature is indicated by the height of the mercury, and traditional records in which the amplitude of the sound is represented by the height of a groove For contrast, see digital ANALYSIS Analysis is the branch of mathematics that studies limits and convergence; calculus is a part of analysis ANALYSIS OF VARIANCE Analysis of variance (ANOVA) is a procedure used to test the hypothesis... numbers.) (3) The base of a polygon is one of the sides of the polygon For an example, see triangle The base of a solid figure is one of the faces For examples, see cone, cylinder, prism, pyramid BASIC FEASIBLE SOLUTION A basic feasible solution for a linear programming problem is a solution that satisfies the constraints of the problem where the number of nonzero variables equals the number of constraints... which you know the lengths of two sides of a triangle and you know one of the angles (other than the angle between the two sides of known lengths) If the known angle is less than 90Њ, it may not be possible to solve for the length of the third side or for the sizes of the other two angles In figure 4, side AB of the upper triangle is the same length as side DE of the lower triangle, side AC is the same... degree measure of an arc and r is the radius, then the length of the arc is 2pru/360 For picture, see central angle The term arc is also used for a portion of any curve (See also arc length; spherical trigonometry.) ARC LENGTH The length of an arc of a curve can be found with integration Let ds represent a very small segment of the arc, and let dx and dy represent the x and y components of the arc (See... figures The area of any polygon can be found by breaking the polygon up into many triangles The areas of curved figures can often be found by the process of integration (See calculus.) ARGUMENT (1) The argument of a function is the independent variable that is put into the function In the expression sin x, x is the argument of the sine function (2) In logic an argument is a sequence of sentences (called... means of algebraic equations Analytic geometry is based on the fact that there is a oneto-one correspondence between the set of real numbers and the set of points on a number line Any point in a plane can be described by an ordered pair of numbers (x, y) (See Cartesian coordinates.) The graph of an equation in two variables is the set of all points in the plane that are represented by an ordered pair of. .. travels from one medium to another, such as from air to water or glass ANGLE OF INCLINATION The angle of inclination of a line with slope m is arctan m, which is the angle the line makes with the x-axis ANGLE OF REFLECTION See angle of incidence ANGLE OF REFRACTION See Snell’s law ANTECEDENT The antecedent is the “if” part of an “if/then” statement For example, in the statement “If he likes pizza, then... altitude of a plane figure is the distance from one side, called the base, to the farthest point The altitude of a solid is the distance from the plane containing the base to the highest point in the solid In figure 3, the dotted lines show the altitude of a triangle, of a parallelogram, and of a cylinder AMBIGUOUS CASE The term “ambiguous case” refers to a situation in which you know the lengths of two ... Dictionary of Mathematics Terms Third Edition Dictionary of Mathematics Terms Third Edition Douglas Downing, Ph.D School of Business and Economics Seattle Pacific... of a triangle, of a parallelogram, and of a cylinder AMBIGUOUS CASE The term “ambiguous case” refers to a situation in which you know the lengths of two sides of a triangle and you know one of. .. glass ANGLE OF INCLINATION The angle of inclination of a line with slope m is arctan m, which is the angle the line makes with the x-axis ANGLE OF REFLECTION See angle of incidence ANGLE OF REFRACTION